Oct 1 Directional Derivatives

Single-Variable Calculus Derivative

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Recall that the definition of a single-variable calculus derivative is this
$$\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h} = f'(x)$$
What does this mean? When you move a very very small amount in the x-direction, what is the slope? As this increment gets smaller and smaller, you get the slope at that point.

Directional Derivative

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The directional derivative at p in the direction v equals
$$D_vf(p) = \lim_{t \to 0} \dfrac{f(p + tv) - f(p)}{t}$$
The magnitude of the vector has to be 1, otherwise we would get different answers in the same direction, which is undesirable.

Understanding this

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Relating it back to $\mathbb{R}$, we get
$$\lim_{t \to 0} \dfrac{f(p + t*1) - f(p)}{t}$$
More generally, imagine you're moving in the direction $<1,1>$.What is the slope in that specific direction? That is the directional derivative.

Partial Derivative

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The partial derivative is a special case of the directional derivative. These are just directional derivatives in the direction of the basis vectors. They are written like this
$$\dfrac{\partial f}{\partial x_n}(p) = D_{e_n}f(p)$$
These are easy to calculate.
Take this for example
$\dfrac{\partial f}{\partial x} x^2y + y^2x$. In this example, you could plug it into the directional derivative equation, which would be
$$\lim_{t \to 0}\dfrac{(x+t)^2y + y^2(x+t) - x^2y - y^2x}{t} = \dfrac{x^2y - x^2y + 2txy + t^2y + xy^2 - xy^2 + ty^2}{t}$$
$$=\lim_{t \to 0}2xy + y^2 + ty = \boxed{2xy + y^2}$$
That was annoying. Now, how you would actually do it is consider y as a constant. For this example, we can rename $y$ to $c$ to make it clearer and do a normal derivative.
$$\dfrac{d}{dx}x^2c + c^2x = 2xc + c^2 = 2xy + y^2$$

Computing Directional Derivatives

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It turns out that we have this equation:
$$D_vf(p) = Df(p)(v)$$
In other words, the directional derivative in the direction of v equals the total derivative at p of the vector.
Therefore, we have an easy way of calculating this.
$$D_vf(p) = Df(p)(v) = \begin{bmatrix}\dfrac{\partial f}{\partial x_1}(p) & \dfrac{\partial f}{\partial x_2}(p) & ... & \dfrac{\partial f}{\partial x_n}(p)\end{bmatrix}\begin{bmatrix}v_1 \\ v_2 \\ ... \\ v_n\end{bmatrix}$$
$$= \dfrac{\partial f}{\partial x_1}(p) * v_1 + \dfrac{\partial f}{\partial x_2}(p) * v_2 + ... + \dfrac{\partial f}{\partial x_n}(p) * v_n$$
Because partial derivatives are fairly easy to calculate, this means you don't have to use the limit definition whenever you want to find a directional derivative.